Calculation of prestressed anchor segment by 3D infinite element

Abstract

Based on 1D infinite element theory, the coordinate transformation and shape function of 3D point-radiation 4-node infinite elements were derived. They were coupled with 8-node finite elements to compute the compressive deformation of the prestressed anchor segment. The results indicate that when the prestressed force acts on the anchor segment, the stresses and displacements in the rock around the anchor segment are concentrated in the zone center with the anchor axis and are subjected to exponential decay. Therefore, the stresses and the displacement spindles are formed. The calculation results of the infinite element are close to the theoretical results.

Keywords

infinite element / prestressed anchor / couple / finite element

Introduction

Various numerical methods have been widely applied in the design and calculation analysis of anchors, in which the finite element method is used frequently because it is accurate in resolving the finite field issue. To resolve the infinite field issue, the mode of a large enough finite field with artificial boundary is adopted to simulate the remote effect of infinite field. The large simulated finite field increases calculation work, and the infinite boundary cannot be reflected well. But the precision of this method is not high [1,2]. The finite element method is extended by the infinite element method to the analysis of infinite field. Therefore, the infinite element method reflects the boundary condition that the displacement of infinite distance is zero, and it can reflect the stresses in the anchor well.

The infinite element method has been developed from 1D to 3D [3-5], from single-way mapping to multi-way mapping [6-9]. The 3D point-radiation 4-node infinite elements were coupled with 8-node, and the stresses of anchor head and segment were analyzed in this paper.

Theoretical derivations

Introduction of radiation functions of infinite element

A new infinite element was proposed in Ref. [5]. In the one-dimension case, the coordinate transformation was given as

x=N_{\mathrm{0}}(\xi )x_{\mathrm{0}}+N_{\mathrm{2}}(\xi )x_{\mathrm{2}},

where N_{\mathrm{0}}(\xi )=-\xi /(\mathrm{1}-\xi ), N_{\mathrm{2}}(\xi )=\xi /(\mathrm{1}-\xi )+\mathrm{1}. As ξ＝+1, both N₀(ξ) and N₂(ξ) have singularity.

Assuming x₁=(x₀+x₂)/2, it can be validated as ξ→1, x→∞; ξ＝0, x=x₂; ξ＝-1, x= x₁, namely, ζ =-1, 0, 1 of the local coordinates correspond to x₁, x_2, and ∞ of integer coordinates respectively (see Fig.1).

It can be seen that the previous radiation takes x₀ as the mapping pole and has two nodes x₁, x₂. Because x₁, x₂ are not mutually independent, assuming x₀＝0, nothing but x₁ of the radiation from the local dimension to the integer dimension can be true, namely single node radiation. The dimension mapping function of single node radiation can be written as follows:

(1)x=N(\xi )x_{\mathrm{1}}=\mathrm{2}{x}_{\mathrm{1}}/(\mathrm{1}-\xi ).

Then ζ =-1, 0, 1 of the local coordinates correspond to x₁, 2x₁ and ∞ of the integer coordinates respectively (see Fig.2).

Mapping function of 4-node point-radiation infinite element

Figure 3 shows the 3D radiation extending from the previous 1D radiation. The 4-node 3D infinite element of the local coordinates can be regarded as four single node radiations of the integer coordinates. All radiation poles are superimposed on each other at the origin. To handle the semi-infinite body issue, the integer coordinates system is formed by taking the boundary center of the semi-infinite body as the origin, and the semi-round body with the origin as the spherical center. The part in the round face is dispersed by the 8-node finite elements, and the other outside the round face is dispersed by the 4-node infinite elements. The radiation pole of the infinite element is the spherical center. The nodes are on the spherical face. The local coordinates extend to infinity. The coordinate radiation is as follows:

\begin{array}{lll}x={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{M}_i{x}_i},& y={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{M}_i{y}_i},& z={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{M}_i{z}_i}.\end{array}

The ζ direction of 3D is defined infinitely. The coordinates of a random point in the ζ＝ζ₀ cover are assumed by the interpolation of four peaks on the cover. The shape function corresponding to η, ξ is invariable on the cover. Because the cover, joined with the finite elements, should continue to satisfy the condition, the interpolating functions of the four nodes on the ζ＝ζ₀ cover can be demonstrated as

(2)\begin{array}{ll}{M}_i=(\mathrm{1}+\xi {\xi }_i)(\mathrm{1}+\eta {\eta }_i)/\mathrm{4},& i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4.}\end{array}

Combining Eqs. (1) and (2), the interpolating functions of the 3D 4-node infinite element are derived as follows:

\begin{array}{ll}{M}_i=(\mathrm{1}+\xi {\xi }_i)(\mathrm{1}+\eta {\eta }_i)/[\mathrm{2}(\mathrm{1}-\zeta )],& i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4.}\end{array}

It is assumed that at the infinite distance the displacement field attenuates from the spherical boundary. The attenuation center is the origin of the integer coordinates. Because all nodes of the point-radiation infinite elements are located in a spherical face, it can be approximately represented as

r_{\mathrm{0}}/r=(\mathrm{1}-\zeta )/\mathrm{2},

where r₀ is the radius of the spherical boundary, r is the distance from the calculation point outside the spherical boundary to the origin. The displacement attenuation functions can be expressed as

(3)\begin{array}{ll}f(\varsigma )={((\mathrm{1}-\zeta )/\mathrm{2})}^{\alpha },& \alpha \ge \mathrm{1.}\end{array}

The displacement transformation formulae are

\begin{array}{lll}u={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{N}_i{u}_i},& v={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{N}_i{v}_i},& w={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{N}_i{w}_i}.\end{array}

where N_i is the shape function N_i={\displaystyle \frac{\left(1+\mathrm{\xi }{\mathrm{\xi }}_i\right)\left(1+\mathrm{\eta }{\mathrm{\eta }}_i\right)}{4}}{\left({\displaystyle \frac{1-\mathrm{\zeta }}{2}}\right)}^{\mathrm{\alpha }}, i=1, 2, 3, 4.

On the basis of the coordinate radiation functions and the displacement transformation functions, the element stiffness matrices of the point-radiation infinite elements can be derived following the process used for the finite elements.

Stress calculation of rock around anchor segment

The prestresses of the prestressed anchor are 2 MN, and the Poisson’s ratio and elasticity modulus ratio are 0.23 and 20 GPa, respectively. It is assumed that the prestresses of the anchor segment are directly exerted on a node of the segment’s initial position. On the basis of symmetry, the quarter model is calculated. The spherical boundary is a spherical face whose radius is 50 m. The element mesh is shown as Fig.4. The finite element method coupled with the infinite element is adopted. The body in the boundary is dispersed by 434 3D 8-node solid finite elements, and the boundary is dispersed by 148 3D 4-node infinite elements. Let α in Eq. (3) of the displacement attenuation function equal to 1.0.

The results are shown as Figs. 5 and 6, in which z begins from the force action spot. Figures 5 and 6 show the results of the infinite element method. Figures 7 and 8 show the results of the infinite element method and the theory derived in Ref. [10]. The following characteristics of the displacements of the rock around the anchor head can be obtained: 1) the displacement of the rock around the anchor segment is mostly located in the region below the pre-stress action spot. The center of the region is the origin. The radius of the region is 2 m, and its depth is 4 m. The displacement above the action spot is similar to that below the action spot. Only the displacements below the action spot are compressed, and the displacements above the action spot are tensed. 2) The attenuation shape of the displacement is the index. Thus, it forms a displacement taper around the anchor head. This indicates that the stress of the rock around the anchor head is tapered. 3) The results from the infinite element method are approximate to the results from the theory derived by the authors. The difference between them is that the curves of the infinite element method rebound. The reason is that these elements are the finite element.

References

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Higher Education Press and Springer-Verlag Berlin Heidelberg